5. Suppose that the joint probability density function of X and Y is given by f(x, y) = C (x² + y), 0 < x < 2, 0 < y < 2 and f(x, y) = 0 otherwise. (a) Find C. (b) Find the marginal density functions of X and Y. (c) Compute P(X > Y). (d) Compute E[X]. (e) Compute E[Y].
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5. Suppose that the joint
and f(x, y) = 0 otherwise.
(a) Find C.
(b) Find the marginal density
(c) Compute P(X > Y).
(d) Compute E[X].
(e) Compute E[Y].
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- The probability density function of X is given by the following table: X 2. 3 4 5 P(x) K 3K 5K 7K 9K 11K 13K Find (i) P (X 0.3 ?7. Let X and Y denote two continuous random variables. Let f(x,y) denote the joint probability density function and fx(x) and fy (y) the marginal probability density functions for X and Y, respectively. Finally let Z = aX + bY, where a and b are non-zero real numbers. (e) Derive an expression for Cov(Z) as a function of Var (X), Var(Y) and Cov(X,Y). [You may use standard results relating to variance and covariance without proof, but these should be clearly stated.]24. Consider a distribution with density on the interval [0, 2]. Let the probability density function (pdf) for this distribution be the following: fy(u) = % on [0, 2] 2 (i) Draw/plot the pdf fy(y) vs. y for the interval [0, 2]. (ii) Determine the Cumulative Distribution Function (CDF), F, (y). (iii) Draw/plot the CDF F,(y) vs y for the interval [0, 2]. (iv) Determine the expected value of the Random Variable (RV) y, i.e., E [y].
- Let X and Y have the joint probability density functionf(x, y) = 5, for 0 < x < y < 1.(i) Find the marginal probability density function of X and Y.(ii) Find the conditional probability density function of X, given Y = y.(iii) Find the conditional mean of X, given Y = y.(iv) Find the conditional variance of X, given Y = y.3. Let X be a continuous random variable. Let f(x) = c(x − 1)³ and Sx = [1,3]. Hint: (x - 1)³ = x³ + 3x − 3x² - 1 (a) What value of c will make f(x) a valid density? (b) What is P(X = 2)? (c) Find E(X). (d) What is P(1 < X < 2)?7. Let X and Y be continuous random variables with joint probability density function given by fxy(xy): Sc, (0, 0 < x < y < 1 otherwise a) Find the value of c. b) Find the marginal distributions of X and Y. c) Find the conditional distribution of Y given X. d) Find the E(Y) and Var(Y).
- c) Let S and T be two independent random variables with probability density functions 1 f(s)= - s > 0, e elsewhere. (e) = t > 0, e 0. elsewhere. Given X = S+ Y and Y = T. Find a joint probability distribution function of X and Y. Then determine the marginal density function of Y.(47) Let X z b(8,-) find E(5+6x) and distribution function.Let X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.
- Let Xand Y be two continuous random variables with joint probability density [3x function given by: f(x.y)%D 0sysxsl elsewhere with E(X) = ECX)- EC) - EC*)= ;and E(XY) = 10 3 E(Y*) = - and E(XY) =; %3D Then the value of the variance of 2X+Y is: O 3/80 O 91/320 43/320 7/20Suppose that X and Y have a joint probability density function given by Sce-32-5y if r, y 2 0 fx.x(r, y) = otherwise (a) Determine the value of the normalization constant c. (b) Find the marginal probability density function fx and state the name of the distribution of X. (c) Find the conditional probability density function fy|x=z•4. Let X, Y be non-negative continuous random variables with probability density functions (pdf) gx(x) and gy (y), respectively. Further, let f(x, y) denote their joint pdf. We say that X and Y are independent if f(x, y) = 9x(x)hy (y) for all x, y ≥ 0. Further, we define the expectation of X to be E[X] = √rg(x)dx, to be the expectation of XY. 0 with a similar definition for Y but g replaced by h and x replaced by y. We also define E[XY] = (0,00)x (0,00) 110,00)x (0,00) 29 (x, y) dedy (0,∞) Use Fubini's theorem (which you may assume holds) to show that if X and Y are independent, then E[XY] = E[X]E[Y]. [2]